A Minimum Distance Bound for 1-Generator Quasi-Cyclic Codes

Let
F_q
be the finite field of
q
elements and
A=F_q[X]/(X^m-1)
be the algebra of
q
-ary polynomials modulo
X^m-1
. The
q
-ary
1
-generator quasi-cyclic (QC) code of block length
ml
, of index a divisor of
l
, with generator
a͟(X) = (a_i(X))_i=0^l-1
is the A-cyclic submodule
V
of
A^l
defined as
Aa͟(X) = {(λ(X)a_i(X))₀^l-1, λ(X) ϵ A}
under the module operation
λ(X)Σ_i=0^l-1a_i(X)Yⁱ = Σ_i=0^l-1㎻(X)a_i(X)Yⁱ
, where
㎻(X) ϵ A
and
㎻(X)a_i(X)
is reduced modulo
X^m-1
. Under the simplifying assumption
(m, q) = 1
, we determine a lower bound on the minimum distance of
V
. The result is achieved by determining a lower bound on the minimal number of not null blocks in a typical codeword of
V
.